Iterative Message Passing Algorithm for Vertex-Disjoint Shortest Paths

As an algorithmic framework, message passing is extremely powerful and has wide applications in the context of different disciplines including communications, coding theory, statistics, signal processing, artificial intelligence and combinatorial optimization. In this paper, we investigate the perfo...

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Bibliographic Details
Published in:IEEE transactions on information theory Vol. 68; no. 6; pp. 3870 - 3878
Main Authors: Dai, Guowei, Guo, Longkun, Gutin, Gregory, Zhang, Xiaoyan, Zhang, Zan-Bo
Format: Journal Article
Language:English
Published: New York IEEE 01.06.2022
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:
Algorithms
Artificial intelligence
Belief propagation
Combinatorial analysis
Convergence
Directed graphs
Electronic mail
Graph theory
Graphs
Heuristic algorithms
Inference algorithms
iterative algorithms
Iterative methods
Message passing
message-passing algorithm
Optimization
Shortest-path problems
Signal processing
Signal processing algorithms
vertex-disjoint shortest path
ISSN:0018-9448, 1557-9654
Online Access:Get full text
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Summary:As an algorithmic framework, message passing is extremely powerful and has wide applications in the context of different disciplines including communications, coding theory, statistics, signal processing, artificial intelligence and combinatorial optimization. In this paper, we investigate the performance of a message-passing algorithm called min-sum belief propagation (BP) for the vertex-disjoint shortest <inline-formula> <tex-math notation="LaTeX">k </tex-math></inline-formula>-path problem (<inline-formula> <tex-math notation="LaTeX">k </tex-math></inline-formula>-VDSP) on weighted directed graphs, and derive the iterative message-passing update rules. As the main result of this paper, we prove that for a weighted directed graph <inline-formula> <tex-math notation="LaTeX">G </tex-math></inline-formula> of order <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula>, BP algorithm converges to the unique optimal solution of <inline-formula> <tex-math notation="LaTeX">k </tex-math></inline-formula>-VDSP on <inline-formula> <tex-math notation="LaTeX">G </tex-math></inline-formula> within <inline-formula> <tex-math notation="LaTeX">O(n^{2}w_{max}) </tex-math></inline-formula> iterations, provided that the weight <inline-formula> <tex-math notation="LaTeX">w_{e} </tex-math></inline-formula> is nonnegative integral for each arc <inline-formula> <tex-math notation="LaTeX">e\in E(G) </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">w_{max}=\max \{w_{e}: e\in E(G)\} </tex-math></inline-formula>. To the best of our knowledge, this is the first instance where BP algorithm is proved correct for NP-hard problems. Additionally, we establish the extensions of <inline-formula> <tex-math notation="LaTeX">k </tex-math></inline-formula>-VDSP to the case of multiple sources or sinks.
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ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2022.3145232